Non-stationary model selection
D. Greenberg, C. Wor, C. Holt, & B. Connors
Sys.Date()
1 Overview
This document contains additional figures exploring the characteristics of the spawner-recruitment relationships for 267 Pacific salmon stocks located throughout the Northeast Pacific region.
2 Spawner-recruit time-series overview
The goal of this document is to examine model performance of various stationary and non-stationary spawner-recruitment model forms for 267 Pacific salmon spawner-recruit time-series spanning five species of anadramous Pacific salmon (Chinook, Chum, Coho, Pink, and Sockeye) spanning from Oregon to Alaska.
2.1 Map of timeseries
## Scale on map varies by more than 10%, scale bar may be inaccurate
2.2 States/Provinces
2.3 Ocean basins
2.4 Series length
#Models
We fit three classes of spawner-recruitment models with 8 models in total, all building off the Ricker stock-recruitment curve:
- Stationary models (Time-Invariant)
- Simple (model 1):
\(log(R_t/S_t) = \alpha - \beta*S_t + w_t\)
\(w_t \sim N(0,\sigma)\)
- AR(1) (model 2):
\(log(R_t/S_t) = \alpha - \beta*S_t + w_t\)
\(w_t = \epsilon_t + \rho*\epsilon_{t-1}\)
\(\epsilon_t \sim N(0,\sigma)\)
- Simple (model 1):
- Dynamic linear models (Time-Varying)
Dynamic \(\alpha\) (model 3):
\(log(R_t/S_t) = \alpha_t - \beta*S_t + w_t\)
\(\alpha_t = \alpha_{t-1} + v_{t}\)
\(v_t \sim N(0,\sigma_{\alpha})\)Dynamic \(\beta\) (model 4):
\(log(R_t/S_t) = \alpha - \beta_t*S_t + w_t\)
\(\beta_t = \beta_{t-1} + u_{t}\)
\(u_t \sim N(0,\sigma_{\beta})\)Dynamic \(\alpha\) and \(\beta\) (model 5):
\(log(R_t/S_t) = \alpha_t - \beta_t*S_t + w_t\)
\(\alpha_t = \alpha_{t-1} + v_{t}\)
\(\beta_t = \beta_{t-1} + u_{t}\)
\(v_t \sim N(0,\sigma_{\alpha})\)
\(u_t \sim N(0,\sigma_{\beta})\)
- Regime shift (Hidden Markov) models (Time-Varying)
- \(\alpha\) Regime (model 6):
\(log(R_t/S_t) = \alpha_k - \beta*S_t + w_t\) - \(\beta\) Regime (model 7):
\(log(R_t/S_t) = \alpha - \beta_k*S_t + w_t\) - \(\alpha\) and \(\beta\) Regime (model 8):
\(log(R_t/S_t) = \alpha_k - \beta_k*S_t + w_t\)
- \(\alpha\) Regime (model 6):
#Stock Characteristics
Here we include an overview of parameters estimated for each stock fit from the aforementioned models.
2.5 Productivity - log(alpha)
Log(\(\alpha\)) estimated from the autocorrelated Ricker function (;model 2) for 267 Pacific salmon stocks. The light histogram is the entire posterior distribution among all stocks, while the dark histogram in the foreground indicates the distribution of median estimates among the 267 stocks:
Summary of the median estimate of log(\(\alpha\)) among all stocks:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## -0.4601 0.8912 1.2149 1.2683 1.5941 3.0508Log(\(\alpha\)) estimated from the autocorrelated Ricker function (model 2) broken down by species:
2.6 Capacity - Smax
\(Smax\) (1/beta) estimated from the simple Ricker function (model 2) for 267 Pacific salmon stocks:
Smax (1/beta) estimated from the simple Ricker function (model 1) broken down by species:
2.7 Sigma
\(\sigma\) (residual deviance) estimated from the simple Ricker function (model 1) for 267 Pacific salmon stocks:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.1358 0.5969 0.7940 0.8669 1.0383 3.1441\(\sigma\) (residual deviance) broken down by species:
2.8 Autocorrelation (Rho)
\(\rho\), autocorrelation coefficient for residuals from the Ricker function (model 2) for 267 Pacific salmon stocks:
\(\rho\), autocorrelation broken down by species:
#Productivity trends
2.9 All stocks
Time-varying productivity trajectories for each stock, with a 5-year moving average and 95% credible intervals around the average.
2.9.1 Productivity
2.9.2 Scaled Productivity
2.9.3 Umsy
### Scaled Umsy 
2.10 Chinook
2.10.1 Productivity
2.10.1.1 ocean region
2.10.1.2 state
2.10.2 Scaled Productivity
2.10.2.1 ocean region
2.10.2.2 state
2.10.3 Umsy
2.10.3.1 ocean region
2.10.3.2 state
2.10.4 Scaled Umsy
2.10.4.1 ocean region
2.10.4.2 state
2.11 Chum
2.11.1 Productivity
2.11.1.1 ocean region
2.11.1.2 state
2.11.2 Scaled Productivity
2.11.2.1 ocean region
2.11.2.2 state
2.11.3 Umsy
2.11.3.1 ocean region
2.11.3.2 state
2.11.4 Scaled Umsy
2.11.4.1 ocean region
2.11.4.2 state
2.12 Coho
2.12.1 Productivity
2.12.1.1 ocean region
2.12.1.2 state
2.12.2 Scaled Productivity
2.12.2.1 ocean region
2.12.2.2 state
2.12.3 Umsy
2.12.3.1 ocean region
2.12.3.2 state
2.12.4 Scaled Umsy
2.12.4.1 ocean region
2.12.4.2 state
2.13 Pink
2.13.1 Productivity
2.13.1.1 ocean region
2.13.1.2 state
2.13.2 Scaled Productivity
2.13.2.1 ocean region
2.13.2.2 state
2.13.3 Umsy
2.13.3.1 ocean region
2.13.3.2 state
2.13.4 Scaled Umsy
2.13.4.1 ocean region
2.13.4.2 state
2.14 Sockeye
2.14.1 Productivity
2.14.1.1 ocean region
2.14.1.2 state
2.14.2 Scaled Productivity
2.14.2.1 ocean region
2.14.2.2 state
2.14.3 Umsy
2.14.3.1 ocean region
2.14.3.2 state
2.14.4 Scaled Umsy
2.14.4.1 ocean region
2.14.4.2 state
